Multidimensional Stationary Probability Distribution for Interacting Active Particles, Supplemental Material

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1 Multidimensional Stationary Probability istribution for Interacting Active Particles, Supplemental Material Claudio Maggi 1, a Umberto Marini Bettolo Marconi 2, Nicoletta Gnan 3, and Roberto i Leonardo 1,4 1 ipartimento di Fisica, Università di Roma Sapienza, I-00185, Roma, Italy 2 Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri, 62032, Camerino, INFN Perugia, Italy 3 CNR-ISC, UOS Sapienza, P.le A. Moro2, I-00185, Roma, Italy and 4 CNR-IPCF, UOS Roma, ipartimento di Fisica Università Sapienza, I Roma, Italy I. PROOF OF THE MAIN RESULT Let us consider the set of stochastic differential equations: ẋ i = xi Φ + η i (1) where i = 1,.., N, xi = / x i and the η i -s are a set of independent Gaussian stochastic processes with zero mean η j = 0 and exponential time-correlation: η i (t)η j (s) = δ ij e t s / The Gaussian colored-noise component η i is necessarily produced by the Ornstein-Uhlenbec process (oob s theorem) 15]: η i = η i + 1/2 Γ i (2) where the Γ i -s are a set of uncorrelated white-noise processes: Γ i (s)γ j (t) = 2δ ij δ(t s) with zero average Γ i = 0. By combining Eq.s (1) and (2) we find 15, 17]: ẍ i + x i Φ + ( δi ) + x ix Φ ẋ = 1/2 Γ i (3) The Unified Colored Noise Approximation (UCNA) is done by neglecting the term ẍ i in Eq. (4). With this simplification we can invert Eq. (4) to obtain ẋ i = M 1 ] i x Φ + 1/2 M 1 ] i Γ (4) where we have defined the matrix M ij = δ ij / + xix j Φ (which is symmetric by construction being xix Φ the Hessian matrix) and its inverse M 1 ] ij. The Foer-Planc equation associated with the (Stratonovich) Langevin equation (4) is given by 18]: a Electronic address: claudio.maggi@roma1.infn.it

2 2 Ω = l,i { } 1 xl M 1 ] li ( xi Φ) Ω + { 2 xl M 1 ( ] li xj M 1 ] ij Ω )} (5) l,i,j where Ω = Ω(x 1,..., x N ) is the probability density function. The stationary solution with vanishing current is found from Eq. (5) by setting: M 1 ] li ( xi Φ)Ω + i M 1 { ] li xj M 1 ] ji Ω } = 0 (6) i,j By multiplying Eq. (6) by M l and summing over l we get: ( x Φ)Ω Ω j xj M 1 ] j = j M 1 ] j xj Ω (7) where we have used the identity l M lm 1 ] li = δ i. We reiterate this step by multiplying by M n and summing over both sides of Eq. (7): Ω M n ( x Φ) + Ω M 1 ] j ( xj M 1 ] n ) = xn Ω (8) j Note that the second term on the l.h.s of Eq. (8) changes sign because xj ( M 1 ] j M n ) = xj (δ jn ) = 0 that gives M 1 ] j ( xj M n ) = ( xj M 1 ] j )M n The second term on l.h.s of Eq. (8) can be further transformed if we use the identity xj M n = xn M j, which stems from the fact that M ij contains the 2 nd -order derivatives of the potential,: M 1 ] j ( xj M 1 ] n ) = j j M 1 ] j ( xn M 1 ] j ) = 1 M x n M where in the last equality M is the determinant of the matrix M and we have used the Jacobi s formula (1/ M ) x M = Tr ( M 1 x M ) 19]. Returning to Eq. (8) we thus have Ω ( ) M n ( x Φ) + 1 M x n M = xn Ω (9) which can be integrated and gives, up to a constant multiplicative (normalization) factor : Ω = exp Φ 2 ] ( xi Φ) 2 M (10) where M is the absolute value of the determinant of M. This last equation which coincides with our main result. i

3 3 II. HAR REPULSIVE POTENTIAL IN 1d: THE GCN CASE For d = 1 the formula (10) reduces to the nown form 15] Ω(x) = exp Φ(x) ] 2 Φ (x) Φ (x) (11) To derive the form taen by Eq. (11) in presence of a steep repulsive potential we consider the fact that, in presence of such a potential, the Ω(x) strongly peas around x = x where Φ (x ) /. This is intuitive since, at the point x, the GCN root mean-squared velocity η 2 = / is balanced by the external (deterministic) velocity component and the particle stops. We further assume that the potential is well approximated by an exponential in the vicinity of x = x Φ λ exp(x x )/λ] (12) which consistently gives Φ (x ) = /. If Ω(x) is peaed around x this must be a maximum which gives the condition 15] Φ (x ) = Φ (x ) (Φ (x ) + 1) 2 If we use Eq.s (12) and (13) into Eq. (11) and we expand it to second order around x we get (13) Ω(x) expa + B(x x ) 2 ] (14) where A = λ + log λ B = 7λ2 + 8λ λ3 2λ 3 + 2λ 2 2 whose integral is Ω(x)dx = 2π exp λ 1 ] ( ) 3/2 + λ 2 7λ 2 (15) + 8λ + 2() 3/2 + 2λ 3 In the limit of λ 0 the potential (12) approaches the potential generated by a hard wall located at x : Φ(x) = if x > x and Φ(x) = 0 if x < x. By taing this limit the integral (15) becomes (note that π/e 1.07): Ω(x)dx = π e (16) Let us now consider one single hard wall of thicness 2x centered at x = 0. We also consider one GCN-driven particle that is free to move around the wall in a 1d space with periodic boundaries located at ±L/2. Using the result given above in this case Eq. (11) becomes Ω = (x x ) + (x + x ) + 1 where (x x ) = 2 δ(x x ), since, in the λ 0 limit, Φ 0 and Ω 1 whenever x > x and Ω 0 if x < x. By integrating Ω we get the normalization constant L/2 L/2 dxω(x) = 2 + L and the normalized probability

4 4 P (x) = (x x ) + (x + x ) + 1 L + 2 (17) where L = L 2x. Eq. (17) can be used to compute the quantity Φ if we recall that Φ (±x ) = ± / and Φ (x) = 0 at any point x ±x. By integrating Eq. (17) we get: Φ = L/2 L/2 dxp (x) Φ 2 (x) = L + 2 (18) III. HAR REPULSIVE POTENTIAL IN 1d: THE RNT CASE To show the similarity between the GCN model and run and tumble (RnT) model in presece of a hard potential we consider one single RnT particle moving in 1d. In the RnT dynamics the noise consists in a two-state Poissonian process happening at rate λ 15]. The RnT noise (also nown as telegraphic noise) is exponentially correlated in time as the GNC, but bimodally distributed switching between two velocities only: η = ±v. The speed v and the characteristic time = (2λ) 1 define the diffusivity = v 2 of the RnT particle in absence of any external potential 6, 11]. We consider the continuity equations that can be written for the RnT model 5]: L(x) = x {L(x) v Φ (x)]} λ 2 L(x) + λ R(x) (19) 2 Ṙ(x) = x {R(x)v Φ (x)]} λ 2 R(x) + λ L(x) (20) 2 where L(x) and R(x) is the fraction of particles with a random velocity component pointing to the left (η = v) and to the right (η = v) respectively. Let us now consider a hard wall as above extending between x and x with periodic boundaries at ±L/2. We consider a stationary solution of Eq.s (19) and (20) of the form L(x) = 2Aδ(x x ) + B/2 (21) R(x) = 2Aδ(x + x ) + B/2 (22) where the constant A represents the fraction of particles going to the left accumulating on the wall at x that is equal to the fraction of particles going to the right accumulating on the wall at x. The constant B in Eq.s (21),(22) represents the fraction of particles that are not in contact with the walls independently on the sign of their velocity. We now focus only on those particle going to the left in the neighbourhood of x (the same conclusions are obtained if we consider those going to the right and x ). Considering the stationary case L = 0 we integrate Eq. (19) it between x and x + ɛ: x +ɛ x +ɛ {L(x + ɛ) v + Φ (x + ɛ)] L(x ) v + Φ (x )]} λ dxl(x) + λ 2 x 2 If we use Eq.s (21),(22) in Eq. (23) we get x dxr(x) = 0 (23) Aλ Φ (0) Φ (ɛ)]b/2 = Aλ + vb/2 = 0 (24) where we have used the fact that Φ (x ) = v. Eq. (24) must be supplemented with the normalization condition L/2 L/2 dx L(x) + R(x)] = 2A + BL = 1 (25) where P (x) = L(x) + R(x) is the overall probability density and L = L x. Using Eq.s (24) and (25) we find B = λ/(v + λl ) and A = v/2(v + λl )] that when expressed as a function of and becomes

5 5 A = B = L L + 2 (26) (27) Using Eq.s (26) and (27) we have P (x) = L(x) + R(x) = (x x ) + (x + x ) + 1 L + 2 (28) (with (x) = 2 δ(x)), which is identical to Eq. (17). IV. RAIALLY SYMMETRIC HAR POTENTIAL Let us consider the case of a spherically symmetric potential for which Eq. (10) becomes Ω(r) = Θ exp Φ(r) ] 2 Φ (r) Φ (r) r + Φ (r) d 1 (29) where Θ is the solid angle in d-dimensions. Let us consider the case d = 2 ( r = x 2 + y 2, Θ = 2π) and a hard potential that is Φ(r) = 0 for r < R and Φ(r) = 0 for r > R. Let us consider one GCN-driven particle that is confined by this circular infinite-well where we expect that Ω(r) peas strongly in correspondence of r = R. We note that Eq. (29) is identical to Eq. (11) multiplied by the term 2π r + Φ (r) when d = 2. We further assume that multiplying Eq. (11) by 2π r + Φ (r) does not change too much the shape of the pea given by the terms where (r) = 2 δ(r). In this way we have that exp Φ ] 2 Φ Φ (r R ) + 1 Ω(r) 2π(r R ) + 1] r + Φ (r) (30) where Φ (R ) = / and Φ (r) = 0 for r < R. The integral of Ω(r) (normalization constant) is computed by noting that, for r < R, Ω(r) = 2πr so that drω(r) = πr 2, while in the neighbourhood of R we get r<r drω(r) = 2π (R + ). Using these results we have from Eq. (30) r R P (r) = 2π(r R ) + 1] r + Φ (r) πr 2 + 2π (R + ) (31) From Eq. (31) we can compute: R Φ = drp (r) Φ 2(R + ) (r) = R R 0 (32) V. COMPARISON OF THE GCN MOEL WITH THE RNT MOEL We briefly illustrate here how the run and tumble (RnT) model compares with the GCN model. The numerical results of the RnT model are compared with the theoretical results obtained by the MUCNASP showing quite unexpected analogies.

6 6 GCN model RnT model (a) (b) (a) (b) (c) (d) (c) (d) FIG. 1: (LEFT) GCN model (a) Probability density function of the position of one GCN-driven particle in presence of the external potential Φ = x 12 (shaded area). Full lines: simulations results, dashed lines: theory, dashed-dotted line: Boltzmann distribution. The more peaed curve corresponds to = 100 µm 2 /s, = 1 s and the less peaed one to = 0.4 µm 2 /s, = 0.1 s (zoomed in the inset) (b) Average value of Φ as a function of for three different values of = 0.1, 0.325, 1 s from top to bottom respectively. Points: simulations results, full lines: theory, dashed lines: theory in the limit of an hard potential, dashed dotted line: equilibrium case with a hard potential. (c) Probability density function of the distance between two GCN-driven particles interacting via the potential Φ = x 12, same legend as Fig. (a). (d) Average value of Φ as a function of for three different values of = 0.1, 0.325, 1 s from top to bottom respectively, same legend as as Fig. (b). (RIGHT) (a) RnT model Probability density function of the position of one RnT particle in presence of the external potential Φ = x 12. Full lines: simulations results, dashed lines: exact theory for the RnT (Eq. (33)). The more peaed curve corresponds to = 100 µm 2 /s, = 1 s and the less peaed one to = 0.4 µm 2 /s, = 0.1 s (b) Average value of Φ as a function of for three different values of = 0.1, 0.325, 1 s from top to bottom respectively. Points: simulations results, full lines: theory (MUCNA), dashed lines: theory in the limit of an hard potential. (c) Probability density function of the distance between two RnT particles interacting via the potential Φ = x 12, same legend as Fig. (a). (d) Average value of Φ as a function of for three different values of = 0.1, 0.325, 1 s from top to bottom respectively, same legend as as Fig. (b). A. One degree of freedom Let us start by the case in which one single particle moves in a 1d space of extension L (periodic boundaries are located at ±L/2) in presence of the external potential Φ = x 12. In this case the probability distribution of the GCN driven particle peas in correspondence of x 1 as shown in Fig. 1(a)(left). This is true also for the RnT model as see Fig. 1(a)(right) where a log-scale has been used for clarity. Note that in this simple case, with one single degree of freedom, the exact P (x) is now for the RnT model 6, 11, 15]: P (x) = A x Φ (x) 2 exp Φ ] (y) dy Φ (y) 2 (33) where A is a normalization constant that absorbs the lower integration limit. Eq. (33) is plotted in Fig. 1(a)(right) as dashed line and it overlaps perfectly the numerical P (x). The average value of the absolute external velocity component Φ can be computed for the GCN case by using the UCNA (Eq. (11)) and is shown as full lines in Fig. 1(b)(left). However we find that this theoretical prediction for the GCN approximates well also the results of the RnT model as shown in Fig. 1(b)(right). This is somehow expected since the expression of the P (x) is almost identical in the two models in the limit of a infinitely steep potential (see Eq.s (17) and (28)) represented by the dashed lines in Fig. 1(b)(left and right).

7 7 GCN model RnT model (a) (b) (a) (b) FIG. 2: (LEFT) GCN model (a) Probability density function of the radial coordinate of one GCN-driven particle in presence of the spherically symmetric external potential Φ = (r R) 12 in 2d (shaded area). Full lines: simulations results, dashed lines: theory, dashed-dotted line: Boltzmann distribution. The more peaed curve corresponds to = 100 µm 2 /s, = 1 s and the less peaed one to = 0.4 µm 2 /s, = 0.1 s (zoomed in the inset) (b) Average value of Φ as a function of for three different values of = 0.1, 0.325, 1 s from top to bottom respectively. Points: simulations results, full lines: theory, dashed lines: theory in the limit of an hard potential, dashed dotted line: equilibrium case with a hard potential. (RIGHT) RnT model (a) Probability density function of the radial coordinate of one RnT particle in presence of the spherically symmetric external potential Φ = (r R) 12 in 2d. Full lines: simulations results, dashed lines: theory. The more peaed curve corresponds to = 100 µm 2 /s, = 1 s and the less peaed one to = 0.4 µm 2 /s, = 0.1 s (b) Average value of Φ as a function of for three different values of = 0.1, 0.325, 1 s from top to bottom respectively. Points: simulations results, full lines: theory (MUCNASP), dashed lines: theory in the limit of an hard potential. B. Two particles in 1d We now consider two particles with coordinates x 1 and x 2 moving in 1d interacting via the pair potential Φ( x) = (x 1 x 2 ) 12 with periodic boundaries at ±L/2. In this case the GCN-driven particles are frequently found in contact leading to the a very peaed P ( x) which is well described by the MUCNASP: Ω( x) = exp Φ( x) 2 ] 2 Φ ( x) Φ ( x) (34) as shown in Fig. 1(c)(left). Eq. (34) allows also to compute the average Φ that is compared with simulation results in Fig. 1(d)(left). ata compare well also with the case of a hard potential obtained by setting 2 in Eq. (18) (dashed lines in Fig. 1(d)(left)) Two RnT particles interacting via the same potential show a more complicated P ( x) that can be obtained numerically but not analytically. This is reported in Fig. 1(c)(right) where P ( x) show a clear secondary pea. This is qualitatively understood as follows: initially the two particle collide with opposite velocities giving rise to the pea at contact, subsequently one of the particles tumble and accelerate for a short distance (being pushed away by the potential Φ = x 12 ), when this becomes negligible the particle has moved at the distance where the 2nd pea is found: in this configuration two particles proceed together in the same direction. We note however that, despite the complicate form of the P ( x), the value of Φ obtained numerically for the RnT model is well approximated by the MUCNASP as shown in Fig. 1(d)(right). This is surprising since the P ( x) of the two models have a quite different form. C. Radially symmetric potential We consider one particle confined in a 2d circular potential well of the form Φ(r) = (r R) 12 where r = x 2 + y 2 and R = 5 µm. The probability density of the GCN-driven particle peas strongly at R R 1 and this is well captured by the MUCNASP (Eq. (29)) as shown in Fig. 2(a)(left). Also the value of the radial external velocity component Φ obtained numerically is well described by the theory (full lines in Fig. 2(b)(left)) and it is close to the hard potential case given by Eq. (32) (dashed lines in Fig. 2(b)(left)). Generalizing the RnT model in 2d we assume that the particle moves at constant speed v and the tumble randomizes completely the swimming direction (see Ref. 27]). The tumble is a Poisson process happening at rate 1 and the axial diffusion coefficient is = v 2 /2 in absence of an external potential. The P (r) of RnT particles interacting with

8 the potential Φ(r) = (r R) 12 is shown in Fig. 2(a)(right) where we can see how the particles accumulate onto the repulsive portion of the potential. This P (r) cannot be predicted theoretically and it is quite different from the P (r) of the GCN-driven particles (see Fig. 2(a)(left)). From the numerical simulations we obtain the quantity Φ that is reported in Fig. 2(b)(right). Again, quite unexpectedly, the values of Φ are close to those found theoretically by the MUCNASP (full lines in Fig. 2(b)(right)) and by the same theory in the hard potential limit (dashed lines in Fig. 2(b)(right)). 8

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